Non-equilibrium Steady States in Catalysis, Molecular Motors, and Supramolecular Materials: Why Networks and Language Matter

All chemists are familiar with the idea that, at equilibrium steady state, the relative concentrations of species present in a system are predicted by the corresponding equilibrium constants, which are related to the free energy differences between the system components. There is also no net flux between species, no matter how complicated the reaction network. Achieving and harnessing non-equilibrium steady states, by coupling a reaction network to a second spontaneous chemical process, has been the subject of work in several disciplines, including the operation of molecular motors, the assembly of supramolecular materials, and strategies in enantioselective catalysis. We juxtapose these linked fields to highlight their common features and challenges as well as some common misconceptions that may be serving to stymie progress.


General comments
The purpose of this document is to provide derivations of equations used and support statements made in the main text. We note that the vast majority of what follows is based on previous work from Astumian and others and this will be cited where relevant. Throughout we will use standard notation used by chemists when discussing reaction kinetics (concentrations, rate constants, equilibrium constants, activation energies etc.), which will hopefully ensure that the equations presented are accessible to the wider chemistry community.

Simple ester hydrolysis networks
1.1a General solution for coupling A→B to X→Y using trajectory thermodynamics 1 At equilibrium steady state, for every molecule of A that is converted to B per unit time, one molecule of B is converted to A. We shall consider how the concentrations of A and B are affected if their exchange is "coupled" to the exchange of X and Y (Scheme S1a), and the concentrations of X and Y are chemostated away from their equilibrium values. To construct the required equations, we consider that all transitions can be coupled (i.e., we are unbiased in deciding which steps are chemically feasible), which provides a new set of potential chemical processes that exchange A and B (Scheme S1b). Scheme S1. a) the reactions to be coupled. b) Processes that can nominally exchange A and B if they are coupled in an unbiased manner.

At steady state,
[ ] = 0, which allows us to write an equation involving the possible transitions in the system: eq. S1 From the form of eq. S1, even without any

1.1b Ester hydrolysis cycle using trajectory thermodynamics
Trajectory thermodynamics 1 requires that, in the first instance, we simply "couple" the MeI hydrolysis reaction to the ester hydrolysis reaction (Scheme S2) in an unbiased manner to generate two additional pathways for ester/carboxylate exchange: Scheme S2. a) The reactions to be coupled. b) New processes that can arise if these processes are coupled in an unbiased manner.
We note that coupled reaction 1 looks chemically unrealistic (it appears to be both reactions happening independently) and that, as drawn, coupled reaction 2 has equivalent species (MeOH and − OH) on both sides of the equilibrium. However, at this stage we assume no chemical knowledge to generate suitable equations, which we can then interrogate chemically at the end. Thus, we assume reaction 1 can take place and that − OH and MeOH may be catalytic in reaction 2 (taken into account by raising [ − OH] and [MeOH] to the power n in coupled reaction 2 (n = 0 for non-catalytic role; n = 1 for catalytic role)). Eq. S2 has exactly the same form as eq. S1, and so equivalent requirements must be met for achieving a nonequilibrium steady state: [OH][MeI] ≠ rxn → the concentrations of the species involved in the coupled reaction must not conform to their equilibrium values; the coupled reaction must be spontaneous in either direction.
Of course, because this is a real system, we can recognise that coupled reaction 1 is not chemically realistic (it represents both reactions happening independently) and so +c (and −c ) can be set to 0. Thus, kinetic asymmetry is automatically a feature of this network once the chemical detail is included. We can also recognise the − OH and MeOH are not catalytic in coupled reaction 2 (which represents the reaction of Iwith the ester) and so n = 0, yielding eq. S3: ] eq. S3

1.1c Ester hydrolysis cycle using the chemical network approach
We can follow the same process starting from the chemical network involving ester hydrolysis and ester formation by reaction with MeI (Scheme S3): ] eq. S4 Eq. S4 has exactly the same form as eq. S3 where ′ +c = −2 . Thus, the chemical network approach is equivalent to the trajectory thermodynamics approach. The latter has the advantage of avoiding introducing bias early in the construction of the mathematical model, but the former is more intuitive for a chemist.

1.1d Flux within the simple ester hydrolysis network
The rate of flux in a chemical cycle can be quantified by the ratcheting constant, r0 (eq. S5): 2 0 = rate of forward step 1×rate of forward step 2×… rate of reverse step 1×rate of reverse step 2×… eq. S5 Applying this equation in the ester hydrolysis network yields: From the form of eq. S6, we can see that there will be net flux over the two different transition states that connect RCO2and RCO2Me if the coupled reaction is spontaneous.

1.1e Comparison between the ester hydrolysis network composed of elementary steps (BAl2 mechanism) and the expanded network in which hydrolysis takes place via a tetrahedral intermediate (BAC2 mechanism)
To confirm that including a two-step hydrolysis (Scheme S4b) does not alter the conclusions drawn using the simple one-step pathway (Scheme S4a) we can compare the forms of the ratcheting constant and Scheme S4. Comparison between the networks established when the hydrolysis reaction is (a) single step (BAl2) and (b) two step (BAc2).
It is straightforward to confirm that the form of r0 is identical to that obtained in the simple network (eq. S6): [OH] as a factor from the numerator yields: [MeOH][I] = 1 (i.e., the coupled reaction is at equilibrium) is identical.

Ester hydrolysis network in which RCO2Me→RCO2is coupled to MeI hydrolysis but R`CO2Me is not.
To examine the behaviour of the network in which only one of the ester conformers is in exchange with RCO2 -(Scheme S5) we can evaluate the exchange of RCO2Me and R`CO2Me, and RCO2Me with RCO2 -.
Scheme S5. Reaction network in which only some processes are coupled to the hydrolysis of MeI.

1.2a Is exchange between RCO2Me and R`CO2Me perturbed by the coupled reaction?
If we focus on the exchange of RCO2Me and R`CO2Me, at steady state: Rearranging, we recover the standard expression (eq. S7) for the relative concentrations of the two conformations at equilibrium. Thus, although the overall system can achieve a non-equilibrium steady state (see below), the conformational exchange equilibrium is not affected by the coupled reaction: eq. S9 Since this equation is identical to the expression obtained without the additional conformational exchange (Section S1.1c), it is clear that the ester hydrolysis cycle is unaffected by the conformational exchange. eq. S10

Network where the hydrolysis of both ester conformers is coupled to
Labelling the first bracket from eq. S10 as "A", we can take ] Substituting these expressions for A and B into eq. S10 yields eq. S11: ] eq. S11 Based on eq. S11 we can see that 0 ≠ 1 (i.e., there is net flux around the cycle) if the following conditions are met: (kinetic asymmetry is present).

1.3b Relative concentrations of species in the cyclic ester hydrolysis network at steady state
To derive an expression for | it is convenient to re-express the network (Scheme S7a) to group the different pathways that link RCO2and RCO2Me in two new kinetic constants,  and , with equivalent constants ` and ` linking RCO2and R`CO2Me (Scheme S7b). We note that the graphical form of this network, with an apparent single kinetic coefficient for each step could be misleading and re-emphasise that we are treating all reactions as reversible; grouping the terms in this way simplifies the algebra to come but does not change the form of the network in any way. Also note the different arrows used in the two representations; equilibrium arrows indicate that the forward and back transmission probabilities (rate constants) are bound by microscopic reversibility whereas the simple arrows in (b) indicate that ,  are not.

Scheme S7. (a) Full network in which both ester conformers undergo hydrolysis/formation. (b) The same network re-expressed in terms of the new kinetic constants (at fixed values of [MeOH], [MeI], [OH] and [I])
, , ` and  `.
Using this notation, we can re-write r0 for this network as (eq. S12) as: eq. S12 We can also use this notation to generate simple expressions for ΨΦ`= 0 ΨΦ` and substituting this expression yields eq. S13: eq. S13 Examining the form of the denominator and numerator, we see that: 1) the term in [] is always 1 if r0 = 1, and so 2) Conversely, if r0 ≠ 1 (i.e., coupled reaction is maintained away from equilibrium, kinetic asymmetry is present), the concentrations of R`CO2Me and RCO2Me are predicted to deviate from the values predicted by K3.
This analysis demonstrates that, even though the exchange between RCO2Me and R`CO2Me is not directly coupled to MeI hydrolysis, because it is part of a cyclic network that contains steps that are, the relative concentrations of these species is perturbed at non-equilibrium steady state and so there is net flux between them.

Minor Enantiomer Recycling
Conversion of benzaldehyde (2) to the corresponding acyl cyanohydrin (4) by reaction with acetoyl cyanide (3) can be coupled to the overall hydrolysis of 3 (Scheme S8a), leading to a reaction network capable of achieving a nonequilibrium steady state (Scheme S8b). Furthermore, because 4 is chiral, by introducing stereoselective catalysts for its formation (cat 1) and hydrolysis (cat 2) it is possible to generate a non-equilibrium steady state in which 4 is significantly enantioenriched. Below we work through the steps required to demonstrate these features.
Note: Throughout, the quoted rate constants are actually "observed" values that include the concentration of the catalyst (i.e., +1 = +1(real) [cat 1]). Compound numbers are as in the main text.

How does coupling acyl cyanohydrin formation to acyl cyanide hydrolysis produce a non-equilibrium steady state?
We can use the simple expression for [ ]  ] eq. S14 From eq. S14 we can see that

What conditions must be met to observe an ee in using the MER strategy?
Using eq. S14 we can directly write equivalent expressions for (reasonable as the overall hydrolysis reaction is strongly favourable and such reactions are conducted with an excess of the coupled reaction substrates), eq. S15 simplifies to eq. S16, in keeping with previous statements from Moberg and co-workers: 3 [ ] eq. S16 Where E1 and E2 are the selectivity factors for the acylcyanohydrin forming and hydrolysis catalysts, respectively.
Thus, the two catalysts reinforce one another, although the exact value of

Iterative MER -reinforcement of ee by independent catalysts
If the reactions in the MER cycle are instead run iteratively under effectively irreversible conditions (i.e., very high thermodynamic driving force, short reaction times which means the reverse process is negligible) we can demonstrate how these catalysts reinforce one another.
In step 1, which is the formation of 4 by reaction of 2 with 3 mediated by cat 1, starting from pure 2 the final value  eq. S17 We can also generate an expression (eq. S18) for t in terms of the conversion of 4S, eq. S18 Substituting this value into eq. S17 (note that `+ 2 +2 = +2 +2 = 2 as [H2O] cancels) yields eq. S19: Using the above equations, taking values of E1 and E2 of 10 and setting = 0.8 (i.e., 80% conversion of the minor enantiomer in the second kinetic resolution step), we find that the ratio 2 : (R)-4 : (S)-4 evolves as shown: Scheme S9. Evolution of the reaction mixture composition if the acyl cyanohydrin (step 1) and hydrolysis reactions (step 2) are performed iteratively.

Operation of catenane 6
In this section we will explore how coupling the base-mediated decomposition of FmocCl to mechanical motion in catenane 6 results in continuous net rotation.
Scheme S10. (a) The reactions that are coupled to generate directional motion in 6. (b) The network that results from the coupling of these reactions. (c) Schematic representation of the operation of 6 (FmocCl, CO2, NEt3 and NEt3.HCl are omitted for clarity).

Derivation of the ratchetting constant for catenane 6 2 Note: throughout, [NEt3] is abbreviated to [B] in the interest of space.
Using eq. S5 starting from 6 and moving clockwise: ).
Thus, the motor's behavior depends on both the equilibrium constant for the coupled reaction, which depends on ∆G rxn (the properties of the molecules) and the concentrations of the species involved, which do not; the

Relationship between the free energy change associated with mass action and the directionality of 6
We can simplify eq. S20 by recognising that

eq. S21
This situation corresponds to strong gating and the maximum directionality of the catenane motor.

Relationship between the directionality of 6 and the maximum work that can be performed
If motor 6 is required to do work against a restoring force, the directionality of the motor is modified. The effect of this force can be quite complicated, for example, it may modify the values of the rate constants. 4 However, if we assume these are unchanged, it has previously been shown 2 that r will be modified compared to r0 by a factor of

eq. S22
If we substitute the value obtained for r0 under conditions of strong gating (eq. S21) we find that wmax = .
From the above discussion, it should be obvious that the maximum work is not limited by ∆G rxn . Furthermore, if the work done over one reaction cycle takes place over a distance l, we see that if F > ∆w , the direction of motor is reversed -as expected, the applied force causes the motor to run backwards.

Free energy changes and maximum work of energy ratchets -catenane 11 +
The operation of catenane 11 + takes place by oscillation of the reaction mixture pH from low (hydrazone gate is labile, ammonium station is protonated) to high (disulfide gate is labile, ammonium station is deprotonated to give an amine), which results in two clockwise half turns of the blue ring (as drawn). Below we dissect these steps to demonstrate that the work done by the motor is not related to the free energy changes of protonation and deprotonation.

Maximum work of catenane 11 under conditions of quantitative protonation/deprotonation
If we assume that the protonation/deprotonation and shuttling steps are each essentially quantitative, it should be obvious, because triazolium-11 + is regenerated and no work is being done, that the free energy change over the full cycle is that of reaction AH with B (G(AH+B)) -it is simply the free energy of the acid/base reaction.
Scheme S12. Operation of 11 + under conditions where both protonation and deprotonation are quantitative, as are the shuttling between the triazolium and the ammonium, and the amine and the triazolium.
The overall free energy change of step 1 (Scheme S13), G(step 1) = G(prot) + G(shuttle1), where G(prot) is the free energy of the acid base reaction between the unbound amine and acid AH. Only G(shuttle1) is affected by a restoring force acting against the direction of shuttling. In the absence of a restoring force, G(shuttle1) = G(tri-NH2) = G(NH2) -G(tri) (tri = triazolium), the free energy associated with binding of the ring to the ammonium and triazole (tri) respectively. If we require the system to shuttle against a load such that the w1 work is done shuttling between the triazolium and ammonium stations, the overall free energy change G(shuttle1) = G(tri-NH2) + w1. If w1 = -G(tri-NH2), the free energy change of shuttling = 0 and thus a 50-50 mixture of the two co-conformations will be produced (i.e., the system is working against its stall force). We assign this value as w1(max).
Step 1 in the operation of 11 + broken down into a protonation and shuttling step against load. indicates that this gate is dynamic (opening and closing) under these conditions..
Using the same approach, we can break step 2 down into a deprotonation event (which is assumed to be quantitative) and a shuttling event. The free energy of deprotonation can be broken down into G(deprot), the free energy of deprotonation for the un-encircled ammonium station, and G(NH2-NH) = G(NH) -G(NH2), the difference in binding energy for the ring encircling the amine and ammonium respectively. As before, the deprotonation event is not affected by the restoring force but G(shuttle2) is, which leads to an analogous result as above w2(max) = -G(NH-tri) = G(tri) -G(NH).
Step 2 in the operation of 11 + broken down into a protonation and shuttling step against load. indicates that this gate is dynamic (opening and closing) under these conditions. This very simple treatment under conditions of quantitative protonation/deprotonation demonstrates that the maximum work possible in such systems is a function of the free energy of shuttling, not the free energy associated with protonation/deprotonation.

Maximum work of catenane 11 under conditions of where protonation/deprotonation are not quantitative
The situation described above is clearly extremely inefficient as the free energy change G(AH+B) would be very large compared with the work done. However, reducing the free energy of protonation/deprotonation results in a more complicated mixture of products in steps 1 and 2 (Scheme S15) and a complete discussion lies beyond this manuscript. However, qualitatively, in step 1 it should be obvious that for a weak acid the degree of protonation is enhanced by the binding of the macrocycle to ammonium unit, and hence the degree of protonation is strongly dependent on the restoring force -disfavoring the shuttling in step 1 will also disfavor protonation. Similarly, in step 2, there is a minimum value of G(deprot) required to overcome the additional cost of G(NH2-NH) -if G(deprot) is too small, the macrocycle will remain bound to the ammonium station and no deprotonation or shuttling will take place. Furthermore, any quantitative treatment of such a system would also consider how long each stimulus was applied for -if the pH is varied faster than the rate of shuttling, the motor will never turn even if protonation/deprotonation is taking place. A general quantitative approach for the analysis of such systems taking all these factors into account has been presented by Astumian, which finds that the maximum work such a motor can perform as the conditions are oscillated is a function of the work done on the system by the stimulus. 5